Multiple-target resolution in the main beam of a conical scan radar

ABSTRACT

A method is disclosed for target ray resolution in coherent, conically scanned radar when a target image is present. An envelope detector and frequency discriminator operate on the IF signal. The fundamental and second harmonic components of the conical scan out of the discriminator are used in a computer along with the DC, fundamental, and second harmonic of the envelope out of the detector to solve five equations in five unknowns. That yields the elevation angles to the target and its image. The larger of the two angles is selected as the elevation angle to the target. The first harmonic and the DC component of the envelope are employed to produce an estimate of the target azimuth angle.

i United States Patent 1 m1 3,781,886 Lank et al. a Dec. 25, 1973 MULTIPLE-TARGET RESOLUTION [N THE Primary Examiner-Malcolm F. Hubler MAIN BEAM OF A CONICAL SCAN RADAR Atl0rney-Samuel Lindenberg et al.

. [75] Inventors: Gerald W. Lank; Gerald E. Pollon,

[57] ABSTRACT both of Claremont, Calif.

A method is disclosed for target ray resolution in co- [73] Asslgnee: Technoogy. Servlce. Corporation herent, conically scanned radar when a target image is Santa Momca Cahf' present. An envelope detector and frequency discrimi- [22] Filed: May 22, 1972 nator operate on the IF signal. The fundamental and a as second harmonic components of the conical scan out [:1] Appl' 255520 of the discriminator are used in a computer along with the DC, fundamental, and second harmonic of the en- [52] US. Cl. 343/7.4 velope out of the detector to solve five equations in [51] Int. Cl. G0ls 9/02 five unknowns. That yields the elevation angles to the [58] Field of Search 343/74 target and its image. The larger of the two angles is selected as the elevation angle to the target. The first [56] References Cited harmonic and the DC component of the envelope are UNITED STATES PATENTS employed to produce an estimate of the target azi- 3,697,992 10/1972 Kleptz ..343 7.4x mum angle 10 Claims, 9 Drawing Figures so ENVELOPE DETECTOR l I FREQUENCY mscmmmi l AToR I I WITHOUT D E 4 LlMITER l 29 I i cos zw REFERENCE FREQUENCY I GEN. SOUARER 20 l l c s l 23 SIGNAL M U SCAN MOTOR PROCESSOR H I Q (FIG?) I 26 m }/25 I MOTOR SERVO I Ea l i GEN AMP. l AZIMUTH DRIVE T MOTOR l 1 i 2 2 1 28 m ,2? d m E20 H5 25 I MOTOR SERVO Es COMPUTER I GEN. AMP (TABLE 2) ELEVATION DRIVE MOTOR l PATENTED (153251973 3.781.885

SHEET 1 BF 5 F| |Cl /V/ ANTENNA AxIs w 1' V DIRECT RAY TARGE l l i f" l2 \J \LRE'ATION AXIS IMAGE RAY PLANE 0F DIRECT a IMAGE ELEv. RAYS PERPENDICULAR TO GROUND AXIS (e) ELEV. AXIS e T COMPUTER V Ed 0 L-P Ed E FILTER )u KpEd SIN w t 4O E2 4i ROTAT. $39 Q AXIS FILTER 's SERVO cos W 44 K= 0 FOR F|G.6 AZ|MUTH I =20 FOR FIGS AXIS (Q) L P E'C FILTER E cos 2w I 45 sQuARER I I 43 L-P E I FILTER D 36 37 To SIN w t COMPUTER L @5 FILTER SIGNAL PROCESSOR -SlN2W T 49 L2 J FILTER FIG? PAIENTEDBEB25I975 3.781.886

SHEET 2 BF 5 FIGZ TARGET- IMAGE SEPARATION I ANTENNA REFERENCE AXIS POINTED AT TARGET (20 LOG O' LOSS IN SIGNALTONOISE RATIO IN ESTIMATING TARGET ANGLE (dB) .75 .8 .9 L0 |.I L2 L3 L4 L5 L6 ()4) SQUINT ANGLE (DEG.)

FIGB

TARGET- IMAGE SEPARATION o o S=0 ()J.) SQUINT ANGLE L3 2o \ifi (ZOLOG O LOSS IN SIGNAL-TO-NOISE (dB) (E TARGET WITH RESPECT TO REFERENCE AXIS (DEG-)-- (2O LOG (I LOSS IN SIGNAL-TO-NOISE RATIO IN ESTIMATING TARGET ANGLE (dB)' RATIO IN ESTIMATING TARGET ANGLE (dB) PATENTEB [IECZS I975 sum 3 or 5 FIG.4

TARGET-IMAGE SEPARATION=L5 ANTENNA REFERENCE AXIS POINTED AT TARGET I 0 G -I.6 L 0,75

.75 .e .9 L0 1.1 L2 L3 L4 L5 L6 (11) SQUINT ANGLE (DEGJ OR 6=09 ALL vALuEs LIE ABOVE 30dB TARGET-IMAGE SEPARATION =o.s

ANTENNA REFERENCE AXIS POINTED AT TARGET e |.e= u 015 7 0 Q I I I I I I I I I .75 .8 .9 m LI [.2 L3 L4 L5 L6 ()1) SQUINT ANGLE (DEG) 1 MULTIPLE-TARGET RESOLUTION IN THE MAIN BEAM OF A CONICAL SCAN RADAR BACKGROUND OF THE INVENTION This invention relates to coherent, conically scanned radar, and more particularly to a method for target resolution with a target image present.

A low-flying target may cause echoes to be returned both directly and as an image due to reflections from the ground. These reflections are referred to hereinafter asan image rays. When an image ray is present in the main beam, it can interfere with tracking.

Considering the case where a coherent radar has a conically scanning antenna of either the rotating asymmetrical dipole type or the mutating feed type, the return of the target and the image ray can be modeled as independent of the scanning characteristic on transmission, as when transmission is from a different antenna than that used to receive the return of the target. If the Cramer-Rao bound for the minimum variance, unbiased estimate of the elevation angle to the target would be plotted for the case where the antenna has a Gauss- .ian pattern, the plots would show a loss in angle estimation accuracy as compared to the case where there is no image present. For high signal-to-noise ratio, a maximum likelihood processor will achieve the Cramer- Rao bound, and for sufficiently high signal-to-noise ratio, the Cramer-Rao bound is the greatest lower bound. However, it is desirable to use more conventional processing.

A conventional conical scan radar system can estimate the elevation angle to the target by using the modulation (envelope) of the IF signal due to scanning. The componenet of the envelope at the conical scan frequency is normalized by the component of the envelope at DC in order to obtain the angle estimate. If two targets are present in the main beam the target to be tracked and its ground reflected image this procedure leads to glinting of the angle estimate, and poor accuracy in elevation angle estimation of the target generally results. However, it has been discovered that when a second image target is present, the second harmonic component of the conical-scan envelope contains information concerning the presence of the second target. There is not enough information there to resolve the targets, but if the transmitter is coherent, there will be added information at the output of a frequency discriminator operating on the IF signal. To be specific, it has been discovered that the fundamental and second harmonic components of the conical scan signal out of the discriminator can be used along with the DC, fundamental, and second harmonic of the envelope to obtain estimates of of the angle to the targets. These components can be put into a computer which solves for the angles to the target. In order to perform these computations, the antenna pattern in the main lobe need be specified, as the computer uses the knowledge of the pattern. The computer may then solve n nonlinear equations in n unknowns. The number of equations corresponds to the unknowns of the targets, i.e., angles, amplitudes of signals, and phases of signals. The number of equations also determines the harmonics of the conical scan out of the discriminator and from the envelope detector which must be sent to the computer.

OBJECTS AND SUMMARY OF THE INVENTION The main object of the invention is to provide radar signal processing and computation methods for tracking a low flying target with an image target present. A further object is to accomplish this main object with conventional signal processing techniques.

These and other objects are achieved in a coherent, conical-scan radar system by processing the IF signal from the radar receiver through an envelope detector to obtain an envelope signal E and through a frequency discriminator to obtain a discriminator signal D, where the IF signal is ofthe form A(t) cos [w r 0(1)], and

(n is the IF center frequency in radians per second, the signal E is proportional to A(t) and the signal D is proportional to A(t) d0(t)/dt. In this expression for the IF signal, A is the amplitude of the signal return and w, is the IF center frequency in radians per second. The signal E is squared and low-pass filtered to obtain a DC signal E proportional to the energy in the IF signal envelope. The squared signal E is multiplied by 3 reference signal sin 0),! from a reference signal generator (driven in synchronism with the conical scan of the radar) to obtain'a product signal equal to -E'* sin 01,1, where a), is the radar system scanning rate in radians per second. This product signal is low-pass filtered to obtain a DC signal E proportional to the first harmonic energy of the envelope signal E. The signals E and E may then be processed to obtain an estimate of the target azimuth angle s as the ratio of E to auE where ,u. is the squint (half cone) angle of the radar scan. The squared signal 15 is multiplied by cos 1.0,! and cos 2w,t to obtain two product signals which are separately filtered to obtain two DC signals E and E proportional to the energy of the fundamental and second harmonic components of the envelope signal E. The disciminator signal D is multiplied by the envelope signal E, and the product signal L is in turn multiplied by sin w,t and --sin 2w,t. The resulting products are then low-pass filtered to obtain signals proportional to the amplitudes of the fundamental and second harmonic components of the discriminator signal D. Knowing the antenna pattern in the main lobe of the conical-scan radar system, these signals E E E L and L are employed in a computer to iteratively solve five equations in five unknowns 6 e A,, A and 8 starting from predetermined initial estimates, and each time substituting computed values for the initial estimates until the difference between output values and the estimates is less than predetermined amounts. The largest of the two elevation angles e and e is then selected for transmission to an antenna elevation servo. An antenna azimuth servo continually receives the sig nal e produced by the signal processor to maintain a vertical plane passing through the antenna axis on the target.

The iteratively computed values A, and A are continually squared and summed by the computer and the sum is continually transmitted to an automatic gain control (AGC) circuit of the radar receiver. Different sets of equations must be used in the computer for the different conditions of separate antennas for transmitting and receiving and a common antenna for both transmitting and receiving. The derivation of the equations in both cases is the same once dependence on the scan angle of the amplitude and phase of the signal returned from the target and its image is taken into con- BRIEF DESCRIPTION OF THE DRAWINGS FIG. la illustrates the geometry of a conical scan radar tracking system with both a target and an image ray present, and FIG. lb illustrates the geometry of FIG. 1a in a plane perpendicular to the rotation axis of the radar system.

FIGS. 2 through 5 are graphs of loss in estimation accuracy versus squint angle (u).

FIG. 6 is a block diagram of a conical scan processor in a radar system having separate transmitting and receiving antennas to track a target when an image is present in accordance with the present invention.

FIG. 7 illustrates a block diagram for a signal processor to be used in carrying out the present invention in a radar system with either one antenna for transmitting and receiving, or with separate antennas.

FIG. 8 is a block diagram of a conical scan processor in a radar system having a single antenna for transmitting and receiving.

DESCRIPTION OF THE PREFERRED EMBODIMENTS When the image ray 10 from a low-flying target 11 shown in FIG. la enters the main beam of a conically scanning antenna 12, normal receiver processing can cause erratic tracking due to the image interference. The IF signal in complex notation, when no noise is present, can be assumed to take the form m= t tet i t trami (1) Here, A, is the amplitude of the i'" signal, i.e. the signal returned from the direct ray or its image, is the phase ofthe 1''" signal, and G[ ,(t)] is the one-way voltage gain of the conically scanning antenna at the time I when the angle between the antenna axis 13 and the 1" signal is {,(t).

The transmission characteristics of the transmitter have been absorbed in the A, and d), in Equation (1). It is assumed that the A, and 5, are constant over at least one scan period. Because the antenna is scanning, the A, and (in, even for a nonfading target, are slightly dependent on the scan angle @(t) as it changes in a periodic manner. This slight dependence can be neglected in the present invention if less accuracy can be accepted, or abbreviated if a separate transmitting antenna is used. Accordingly, a separate transmitting antenna is assumed for the present. Also, it is necessary to assume the transmitter is coherent for at least one scan period for the A, and dn to be constant over this time.

The radar has a short-pulsed transmitter so that the S(t) in Equation l) are only observed at discrete times t,,. However, assuming the pulse repetition rate is large compared to the scan period, the received signal can be assumed to have the form of Equation (1) at all r.

The scanning antenna rotates at a fixed rate 0), rad/sec about an axis 14 and the antenna axis 13 makes an angle IJ. with the axis of rotation as shown in FIGS. 1a and lb. This angle is called the squint angle from the geometry of the situation it is evident that for f t!) small (compared to one radian),

In Equation (2), e and e are the angles the 1"" signal makes with the axis of rotation in the elevation and azimuth planes, respectively, as shown in FIG. lb, Equation (2) assumes the antenna axis at t O is in the elevation plane, i.e., that the angle ru t of the antenna axis at time t is measured from the elevation plane as shown in FIGS. 1a and lb. This choice of a reference plane is, however, arbitrary.

Since the two signals correspond to the target and its image ray, it is permissible to assume that the azimuth angle is the same to both. Thus,

where 0 is the total one-way 3-dB beamwidth of the antenna.

It will be assumed that additive white noise is present in the signal S(t) in Equation (1). To achieve the pres- .ent invention, the best accuracy with which the angle to the target can be estimated in an unbiased manner is first obtained. It is assumed that the seven parameters e A 6, A (b and 6,, are unknown. A processor which will come close to achieving this accuracy can then be used to solve for these unknowns, which can be reduced to five, as will be presently understood.

If the signal-to-noise ratio can be assumed good, then realistic bounds on the minimum variance of unbiased estimates of the seven unknown parameters are given by the Cramer-Rao bound. This is obtained as follows. Let a i= 1,2, ,7 be the seven unknown parameters, e A 4),, 682, A (#2, e respectively. Let T be the time of observation of the signal S(t) in Equation (1). It will be assumed that T is an integer number of scan periods. Thus,

where m is a positive integer. Form the 7 X 7 information matrix I with the components 1 Real 1 l I BSU) 6S0) dt 17 T 0 6 H 1.-

5 6 mum variance unbiased estimate of the parameter 6, is 6,, 0, Le, in the case of the reference axis actually in given y mm the elevation plane of the target and its image) the re- Performing the operations in Equation (7), using suits shown in Table l for 1. Equations (1) through (4) and (6), yields (in the case Here the P070, are defined as follows.

PE, (2 463,) all, 8,ue,. u.M, 2u aN 2e, ZLB sin a 2 \GM. sin a Pjl 2 Ze VdL cos 6 2p, vdM cos 8 COS Here, I,,(.) are the modified Bessel functions of the first kind, where From Table I, it is evident that e is uncoupled from all the other parameters because of the zeros in the last row and column. This implies that the variance of the estimate of the common azimuth angle of the direct and image ray is given by In Equation l2), |1,| is the determinant corresponding to the matrix in solid lines, while lM is the determinant formed by eliminating the first column and first row of 1,.

An examination of matrix I, in Table 1 shows that A can be removed as a common factor from each row and column headed by a and (1),, while A can be'removed as a common factor from each row and column headed by 6 and Furthermore, la can be removed as a' common factor from each row and column headed by a and e Let 7, be the matrix formed from 1 in Table l by setting A, A \/cT= l, and let M be the matrix formed by eliminating the first row and column of 1,. Then the above discussion implies l ll l 2l ll l l 014211442 Thus, from Equations (l2), (l3), and (14) the following variance is obtained.

N 15 21rmaA Il If the image were not present, then the variance on the estimate of the elevation position of the target would be found by inverting the 3 X 3 matrix which is in the upper left-hand corner of I, in Table l. The upper left-hand corner of the inverted matrix would then be the minimum variance on an unbiased estimate of e Performing this procedure for the case s (l, the following minimum variance is obtained for when the image is not present ln Equation l 6), the value ofu is labeled 1.1 in order to consider the squint angle o as a parameter to be optimized when the image is present. Thus n will represent a reference squint angle (such as the one which mini- Thusrr' is the minimum variance unbiased estimate of the elevation angle of the target when the image is present, normalized to the minimum variance unbiased estimate of the angle when the image is not present and the target is on the reference (rotation) axis. It can also be thought of as the effective loss in signal-to-noise ratio in estimating the target angle.

By use of a computer, Table l, the definitions of M and I in Equation (17), and Equations (8) and (9), the normalized variance can be found in Equation (17). If the 3-dB beamwidth of the antenna is specified, then Equation (5) can be used to obtain a. This has been done for the case of interest, namely, 6,, 1.6 and 11. 0.75".

Curves have been plotted for this case in FIGS. 2 through 5, inclusive. If all angles on these curves were normalized to the beamwidth 0 the resultant curves wouldhold for any size of antenna.

in FIG. 2 0 is plotted in decibels as a function of u for a target image separation of 1. This is the minimum separation at which the system must meet specifications. The results are a function of 8, i.e., the electrical path length (phase) difference (qSr between the target and image ray. However, they are symmetric with respect to 0 and i90. Thus they need only be plotted for 0 8 5 90, which has been done in all four FIGS. 2 through 5.

For the majority of cases in FIG. 2, the optimum squint angle to use is in the vicinity of 1.3". Also, in this figure the target wasplaced along the reference axis, i.e., e 0. In FIG. 3 is plotted the estimation accuracy versus the target position with respect to the reference axis for the case p. 1.3. It can be seen that the minimum accuracy loss occurs in the vicinity of the reference axis but is quite flat. A positive value of e in FIG. 3 corresponds to the reference axis lying between the target and its image.

In FIG. 4 are plots analogous to FIG. 2 for a targetimage separation of 1.5". It can be seen that at this larger separation the losses are smaller than for the l case. Similarly, in FIG. 5 are plots for a separation of 0.5". In this case, the losses are larger than for the 1 case.

Next consider a processor to perform the estimation of the target angle when the image is present. For good signal-to-noise ratio, which is the situation of concern, the maximum likelihood detector will achieve the performance indicated in FIGS. 2 to 5. However, in order to use conventional hardware, a different approach is taken in accordance with the present invention.

The IF signal is normally envelope detected in a receiver, so this signal is available. If a frequency discriminator without a limiter is employed for operation on the IF, its output can also be used since the transmitter is coherent. These two signals will have components at DC, w,, and, in general, all the harmonics of w, The

lower harmonics will contain most of the information in the signal, since the higher-order harmonics will contain higher-order terms of the angles to be estimated and will thus be more easily corrupted by noise. Therefore the method of the present invention uses as many of the lower harmonic outputs as are needed to define a set of equations, in the noise-free case, which is sufficient to solve for the unknowns in Equation (1).

The envelope of the signal is obtained from Equation (1) as E= ism lAictexm +Atctt xrne l Us) represented by D, then sin5 Doing this and retaining up to second-order terms in the angles, it is found that six equations needed to solve for the unknowns are obtained from E E E E L,,, and L Furthermore, it has been discovered that Era/nub}; is a good estimate of s, when a separate antenna is used for transmitting and receiving, so that it is used in the equations associated with E E E L and L to solve for the remaining five unknowns A,, A 691, 92, and 5, thus reducing the number of equations needed from five to six.

The processor described above is shown in FIG. 6, while the five equations in five unknowns are contained in Table 2. The larger of the angles 6,, and e at the output of the computer can be taken as the angle to the target, since the target is above the image. A simple comparator can be provided to make that choice. The computer itself may be implemented as an analog computer, or as a digital computer if the input signals are in digital form, or are converted at the input to the computer into digital form. In either case, the implementation shown in FIG. 6 assumes a signal processor 20 implemented in accordance with FIG. 7, and a computer 21 for iteratively carrying out the computations of the equations in Table 2. It is further assumed that the computer includes means for forming the sum A, A for AGC control and a comparator to select the larger of the angles e,., and 6, for the elevation servo control.

TABLE 2.

THE FIVE EQUATIONS IN COMPUTER IN FIVE UNKNOWNS (e ,e ,A,,A ,8)

ANTENNA HAS ONE-WAY VOLTAGE PATTERN C(E) (u 1. IS SQUINT ANGLE OF THE CONICAL SCAN 5,, IS AZIMUTH ANGLE OF DIRECT AND IMAGE RAY e, ,e, ARE ELEVATION ANGLES OF DIRECT AND IMAGE RAY.

The embodiment of FIG. 6 assumes a separate antenna (not shown) for transmission, although it may have only the antenna 22 shown for both transmitting and receiving. A scan motor 23 drives an antenna to rotate its boresight axis about a reference (rotation) axis at the constant rate (0,. It also drives a reference generator 24 to produce two reference signals sin mg and cos for use in the signal processor 20. An output e, from the signal processor drives the rotation axis in azimuth.

through a servo 25 and a motor 26 while the output e from the computer 21 drives the rotation axis in elevation through a servo 27 and motor 28.

The computer receives as inputs the signals e,,, E E E L and L from the signal processor to solve the equations of Table 2 for: A and A, combined in the AGC signal; 2,, and from which the output 6 is selected and 6 used only in the computer. To generate the inputs to the computer, the signal processor 20 receives the reference signals sin w,,t and cos (0,! and through a frequency squarer 29 the reference signals sin 2w,,t and cos 210,1. It also receives signals D and E. The latter is the output from an envelope detector 30 which receives the IF signal from the radar receiver 31 in a conventional manner. A frequency discriminator without limiter 32 receives the IF signal and produces the signal D. The IF signal is of the form A(t) cos [w ,,t 0(t)]. Therefore E is proportional to A(t) while D is proportional to A(t)d 0(t)/dt.

Referring now to FIG. 7, the signal processor receives the signals E and D, and the reference signals, and uses as many of the lower harmonics of the two signals as are needed to define a set of equations sufficient to solve for the unknowns in Equation l Ifthe output E of the detector 30 (FIG. 6) is as given by Equation (I8), the output of a mixer 36 (FIG. 7) is given by Equation (19). A squarer 37 produces the signal E", which with the signal L makes it possible to provide the harmonic series of Equation (21). The output of the squarer is transmitted through a low-pass filter 38 to provide the input E, to the computer 21 of the form represented by the first of the five equations in Table 7 The signal E is multiplied by a signal sin (0,! in a mixer 39, and the product is transmitted through a lowpass filter 41 to produce a signal E which, when divided by 2,uE,, in a divider 41 yields a good estimate of the azimuth error E,,. That error signal is transmitted to the computer 21, as noted hereinbefore. However, it should be noted that since the azimuth error has no image ray interference, it can be computed in the conventional manner, or in any other convenient way.

To complete the inputs to the computer 21, the signal E is multiplied by cos w,t and cos 20),! in respective mixers 42 and 43, and the respective products are transmitted through low-pass filters 44 and'45 to provide the signals E and E of the form represented by the second and third equations of Table 2. Similarly, the signal L from the mixer 36 is multiplied by sin w,t and sin 20),! in mixers 46 and 47, and the respective products are transmitted through low-pass filters 48 and 49 to provide the signals L and L of the form represented by the fourth and fifth equations of Table 2. All six inputs are then used in the computer to solve the equations of Table 2 for the unknowns. Phase inverters (not shown) are used to obtain the negative signs where needed for the sine and cosine reference signals used by the signal processor in producing these input signals to the computer.

In operation the computer of FIG. 6 is programmed to iteratively solve the equations of Table 2 using as initial estimates the following values.

After the first computation, each iteration employs as the initial condition the valuesjust computed. There fore, the values computed during each iteration replace the previous estimates for the next iteration. This process is continued at a fast rate to bring the reference axis on the target. The iteration is completed once the outputs are obtained to the following accuracy.

Referring now to FIG. 8, a second embodiment of a conical scan processor is shown for tracking a target when an image ray is present in a coherent pulsed radar system using a single antenna 50 for both transmitting and receiving radar pulses. A conventional duplexer 51 connects the antenna to a transmitter 52 and isolates the receiver 53 during transmission. Between transmission periods, the receiver accepts return signals and produces an IF signal. As in the case of the first embodiment, the fundamental and second harmonic components of the IF signal can be used along with the DC, fundamental and second harmonic of the envelope to obtain estimates of the angle to the target. These components are processed in a processor 54 and computer 55 to solve for the azimuth and elevation angles to the target. The signal processor is implemented as shown in FIG. 7 for the first embodiment. The difference between the two embodiments is that in this second one the computer is programmed to solve the five equations of Table 3 for the five unknowns E E A A and 8.

3,781,886 13 14 TABLE 3 THE FIVE EQUATIONS 1N COMPUTER lN FIVE UNKNOWNS A t h y Voltage Pattern o having the algebraicly larger angle is selected as the u is squint angle of the conical scan; ea is azimuth real target.

angle of direct and image ray; ,6 are elevation an- As noted hereinbefore, the exact form of the equaglss dlrectahd Image y; n(') is the fied sse tions in the computer will depend upon the antenna function of the first kind of order n. pattern 0(5). The foregoing analysis assumes a Gaush azimuth angle n is Calculated as in ths first sian-shaped pattern given by Equation (4). if the patbodimem through the Signal Procsssof 0f as is tern does not have that shape, but has some other the input A], to the computer 55. The other inputs E known symmetrical shape, the function defining that a i and a to the computer e o late as shape would be substituted into Equations (18) and in the first embodiment through the signal processor of (20) i t ad of that of Equation (4). The harmonic Se- G- Thecalculated azimuth angle 60 is pp to an ries would then be obtained in Equation (21) using the azimuth servo to control an azimuth drive motor. The resulting f ti f Equations (13 d (20) Th difference is how the elevation angle e is calcu' knowns would still be obtained from the six equations lated by the computer 55 to drive an elevation servo 64 associated i h E E E E L and L that controls an elevation drive motor 65. However, wh h transmitting d receiving antenna are the this difference is not one of concept, or of basic princisame (FIG 3) h f f h return signal is (taking P h? h deflvatltm of the five equations in five account of the scanning motion of the transmitter durknowns, but simply one of taking into account the dei transmit) pendence of the amplitude and phase of the signal re- 0 qgy cg gna 2 1;

turned from the target and its image on the scan angle h a i l antenna i d, Substituting this form of S(t) intolEquations (l8) and Another difference illustrated by the set of five equathe envelope E and dissfiminstol' Output L can be tions in Table 3 is the derivation of equations using the found in manner analogous to that used to Obtain Equal exact functions which are the modified Bessel functions tiOhs and for the Case (FIG. Where p I,.(') of the first kind, where n is the orde of he f rate antennas are used for transmitting and receiving. tion from 0 to 2. However, this further difference is not Again assume K6 has a Gaussian shalte as Eqlla' one of concept either since both sets of equations in tiOh A harmonic series as in q n is Tables 2 and 3 are derived in a strictly analogous m performed and the resultant functional forms of the six ner. equations needed to solve for the unknowns are again The received signal may be a CW or pulsed radar sig- Obtained from 4, lsi m m ia! and L28 Furthernal in the first case, but only a pulsed radar signal in the more, it has been discovered in this Case that it/ 1 11 second case (FIG. 8). in the second case, several pulses is a g estimate of 50 that it is used in the q are required from the target during each revolution. In tiOhS associated With li, lct tc L131 and u to solve both cases there is amplitude modulation at the scan for the remaining five unknowns A,, A e e and 8. frequency,and maximum amplitude occuring when the The resulting five equations are those shown in Table scanning antenna beam is closest to the target or the 3. In performing the harmonic series for this case, the

image. What is more important is that in both cases the higher-order rm in he ngle were not dropped. low harmonics of the conical'scan out of a discrimina- This accounts for the appearance 0f the n(') which are or and envelope detector are used to resolve two tarthe modified Bessel functions of order n. if the pattern gets when the signal to noise ratio is high. The target G()" does not have the Gaussian shape as indicated in Equation (4), but has some other known symmetrical shape, then the harmonic series in Equation (21) can still be obtained in the manner just discussed.

What is claimed is: 1. A method for target ray resolution in coherent, conically scanned radar when a target image is present, using a radar with a known symmetrical pattern comprised of:

receiving a radar return signal, and processing said return signal through an envelope detector to obtain a signal E, and through a frequency discriminator to obtain a signal D, where said return signal is of the form A(t) cos [w t 6 (t)] in which form 0),, is the center frequency of said return signal in radians per second, said signal E is proportional to A(t) and said signal D is proportional to A( t) de(t)/dt;

squaring said signal E to obtain a signal E proportional to [A(t)] passing said signal E through a low-pass filter to obtain a DC signal E proportional to the energy in said signal E; multiplying said signal E by said signal D to obtain a product signal L; generating reference signals sin ant and cos m,t in synchronism with the conical scan of said radar, where w, is the radar system scanning rate in radians per second; squaring said reference signals to obtain addition ref erence signals sin and cos 2w,t; multiplying said signal E by said signal sin w ,t,

inverted, to obtain a signal E sin w r; passing said signal -E sin (0,! through a low-pass filter to obtain a signal E proportional to the first harmonic energy of said signal E; using said signals E and E to obtain an estimate of the target azimuth angle 5,, as the ratio of the signal E to E, times a constant K, where u is the halfcone angle of the radar scan; multiplying said signal E by said signal cos ant inverted to obtain a signal I" cos ant and by said signal cos 2am to obtain a signal E cos 2w,,t; passing said signals -E cos w,t and E cos 2 an! through separate low-pass filters to obtain two signals E and E proportional to the energy of the fundamental and second harmonic components, respectively of the signal E; multiplying said signal L by said signal sin co t to obtain a signal L sin w,,t and by said signal sin 2 w,t inverted to obtain a signal -L sin 2 ant; passing said signals L sin w,,t and L sin 2 t through separate low-pass filters to obtain two signals L and L proportional to the amplitudes of the fundamental and second harmonic components, respectively, of said signal D; using a known function of said antenna pattern, and the assumption that the azimuth angle of the target and the target image are equal, programming a computer to use said signals E E E L and L to solve five equations in five unknowns A A a s and 8, where A, and A are the amplitudes of the target, and target image return signals e and e are the elevation angles of the target and target image, and 8 is the phase difference between the target and target image signals; and selecting the largest of said two elevation angles E and e as the elevation angle 6,. of the target. 2. A method as defined in claim 1 wherein said computer is programmed to iteratively solve said five equations in five unknowns starting from initial estimates of the unknowns, and after each iteration substituting computed values of said five unknowns for the initial estimates until the differences between output values and the last estimates have been reduced to predetermined amounts.

3. A method as defined in claim 2 wherein said conically scanned radar is employed for target tracking through azimuth and elevation servos by continually outputting said signal a to said azimuth servo and continually outputting to said elevation servo said largest of said two elevation angles as a signal proportional to 5 when said difference has been reduced to said predetermined amounts.

4. A method as defined in claim 3 wherein said radar is provided with automatic gain control by continually outputting a signal proportional to the sum of the squares of said computed values A, and A when said difference has been reduced to said predetermined amounts.

5. A method as defined in claim 1 wherein said radar receives energy from an antenna separate from an antenna used for transmitting energy and said antenna has a known pattern of symmetrical shape with a function C(5 and wherein said five equations said computer is programmed to solve are derived by using 0) p e e 2pte ,cosm,t 2;.te sinw,t in the following equations 'togetherwith the known function of the antenna pattern to expand E and L from these equations into a harmonic series, i.e N H v V E E 2 (E cos iw t E sin ico t) L L Z (Li cos im,,t L1,, sin tam) L aA A uw,,(e e )sin8 L a A A u w,,(e )sin8 when said antenna'has a Gaussian shape.

7. A method as defined in claim 6 wherein said constant K is equal to a, and a is the same coefficient a in the Gaussian shaped pattern function 8. A method as defined in claim 1 wherein said radar L L L c w t L- s 1 receives from the same antenna used to transmit and d 2 w 05 l s in said antenna has a known pattern of symmetrical shape with a function C(5 andwherein said five equations said computer is programmed to sol are d i d by 5 retaining up to second order terms in the angles to obtam six equations in six unknowns E E E E L usm 592 5a and L and using said signal e as an estimate of target 5 t) 2, 905w, z gg sinw t azimuth in the five equations associated with E E in the following equations E L and L to solve for the remaining five un- L= A ,A G( )[dG( )/d Si s 9. A method as defined in claim 8 wherein said five A A ,G( )[dG( ,)/d i s equations said computer is programmed to solve are as together with the known function of the antenna patfollows:

2 2 -2u --211: Aie ga,, +Age 1 so c 2AM; cos 28c 10 (4M8! 4apte 2 2 gy 1 ,4 6 (ee e8 sin 8 [Iowan/ice 2a,u.e I2 (2a,u.ee 2ap.e

2 2 4azI LzwsA1A2e 1 e (e e sin 5[I,(2ape +2apee -I (2a1.te +2ap.e

tern to expand E and L from these equations into a 'when said antenna has a Gaussian shape. harmonic series, i.e., 40 10. A method as defined in claim 9 wherein said constant K is equal to 20 where a is the same coefficient a as in the Gaussian shaped pattern function (E2 =12 i cos iw t Ei sin iw t) G ifl i 

1. A method for target ray resolution in coherent, conically scanned radar when a target image is present, using a radar with a known symmetrical pattern comprised of: receiving a radar return signal, and processing said return signal through an envelope detector to obtain a signal E, and through a frequency discriminator to obtain a signal D, where said return signal is of the form A(t) cos ( omega ot + theta (t)) in which form omega o is the center frequency of said return signal in radians per second, said signal E is proportional to A(t) and said signal D is proportional to A(t) de(t)/dt; squaring said signal E to obtain a signal E2 proportional to (A(t))2; passing said signal E2 through a low-pass filter to obtain a DC signal Ed proportional to the energy in said signal E; multiplying said signal E by said signal D to obtain a product signal L; generating reference signals sin omega st and cos omega st in synchronism with the conical scan of said radar, where omega s is the radar system scanning rate in radians per second; squaring said reference signals to obtain addition reference signals sin 2 omega st and cos 2 omega st; multiplying said signal E2 by said signal sin omega st, inverted, to obtain a signal -E2 sin omega st; passing said signal -E2 sin omega st through a low-pass filter to obtain a signal E1s proportional to the first harmonic energy of said signal E; using said signals Ed and E1s to obtain an estimate of the target azimuth angle Epsilon a as the ratio of the signal E1s to Mu Ed times a constant K, where Mu is the half-cone angle of the radar scan; multiplying said signal E2 by said signal cos omega st inverted to obtain a signal -E2 cos omega st and by said signal cos 2 omega st to obtain a signal E2 cos 2 omega st; passing said signals -E2 cos omega st and E2 cos 2 omega st through separate low-pass filters to obtain two signals E1c and E2c proportional to the energy of the fundamental and second harmonic components, respectively of the signal E; multiplying said signal L by said signal sin omega st to obtain a signal L sin omega st and by said signal sin 2 omega st inverted to obtain a signal -L sin 2 omega st; passing said signals L sin omega st and -L sin 2 omega st through separate low-pass filters to obtain two signals L1s and L2s proportional to the amplitudes of the fundamental and second harmonic components, respectively, of said signal D; using a known function of said antenna pattern, and the assumption that the azimuth angle of the target and the target image are equal, programming a computer to use said signals Ed, E1c, E2c, L1s and L2s to solve five equations in five unknowns A1, A2, Epsilon 31, Epsilon e2 and delta , where A1 and A2 are the amplitudes of the target, and target image return signals Epsilon e1 and Epsilon e2 are the elevation angles of the target and target image, and delta is the phase difference between the target and target image signals; and selecting the largest of said two elevation angles Epsilon e1, and Epsilon e2 as the elevation angle Epsilon e of the target.
 2. A method as defined in claim 1 wherein said computer is programmed to iteratively solve said five equations in five unknowns starting from initial estimates of the unknowns, and after each iteration substituting computed values of said five unknowns for the initial estimates until the differences between output values and the last estimates have been reduced to predetermined amounts.
 3. A method as defined in claim 2 wherein said conically scanned radar is employed for target tracking through azimuth and elevation servos by continually outputting said signal epsilon a to said azimuth servo and continually outputting to said elevation servo said largest of said two elevation angles as a signal proportional to epsilon e when said difference has been reduced to said predetermined amounts.
 4. A method as defined in claim 3 wherein said radar is provided with automatic gain control by continually outputting a signal proportional to the sum of the squares of said computed values A1 and A2 when said difference has been reduced to said predetermined amounts.
 5. A method as defined in claim 1 wherein said radar receives energy from an antenna separate from an antenna used for transmitting energy and said antenna has a known pattern of symmetrical shape with a function G( xi 2), and wherein said five equations said computer is programmed to solve are derived by using epsilon e1 epsilon e2 epsilon a xi 2i(t) Mu 2 + epsilon 2e + epsilon 2a 2 Mu epsilon e cos omega st - 2 Mu epsilon a sin omega st in the following equations E S(t) A1G( xi 21(t)) + A2G( xi 22(t)) ej L A1A2 xi 22G( xi 21)(dG( xi 22)/d xi 22)sin delta - A1A2 xi 21G( xi 22)(dG( xi 21)/d xi 22) sin delta together with the known function of the antenna pattern to expand E2 and L from these equations into a harmonic series, i.e.,
 6. A method as defined in claim 5 wherein said five equations said computer is programmed to solve are as follows: Ed A21 + A22 + 2A1A2cos delta + epsilon 2aa(4A21 Mu 2a + 2A21 + 4A22 Mu 2a + 2A22 + 8A1A2 Mu 2acos delta + 4A1A2cos delta ) + epsilon 2e a(4A21 Mu 2a + 2A21 + 2A1A2 Mu 2acos delta + 2A1A2cos delta ) + 4A1A2 Mu 2a2 epsilon e epsilon e cos delta + epsilon 2e a(4A22 Mu 2a + 2A22 + 2A1A2 Mu 2acos delta + 2A1A2cos delta ) E1c 2A21 Mu a epsilon e + 2A22 Mu a epsilon e + 2A1A2 Mu acos delta ( epsilon e + epsilon e ) E2c 2A21 Mu 2a2 epsilon 2e + 2A22 Mu 2a2 epsilon 2e + A1A2 Mu 2a2( epsilon e + epsilon e )2cos delta - 2A21 Mu 2a2 epsilon 2a - 2A22 Mu 2a2 epsilon 2a - 4A1A2 Mu 2a2 epsilon 2acos delta L1s aA1A2 Mu omega s( epsilon e - epsilon e )sin delta L2s a2A1A2 Mu 2 omega s( epsilon 2e - epsilon 2e )sin delta when said antenna has a Gaussian shape.
 7. A method as defined in claim 6 wherein said constant K is equal to a, and a is the same coefficient a in the Gaussian shaped pattern function G( xi ) e
 8. A method as defined in claim 1 wherein said radar receives from the same antenna used to transmit and said antenna has a known pattern of symmetrical shape with a function G( xi 2), and wherein said five equations said computer is programmed to solve are derived by using epsilon e1 epsilon e2 epsilon a xi 2i(t) Mu 2 + epsilon 2e + epsilon 2a - 2 Mu epsilon e cos omega st - 2 Mu epsilon a sin omega st in the following equations E (A1G( xi 21(t)) ej + A2G( xi 22(t)) ej )2 L A1A2 xi 22G( xi 21)(dG( xi 22)/d xi 22) sin delta - A1A2 xi 21G( xi 22)(dG( xi 21)/d xi 21) sin delta together with the known function of the antenna pattern to expand E2 and L from these equations into a harmonic series, i.e.,
 9. A method as defined in claim 8 wherein said five equations said computer is programmed to solve are as follows:
 10. A method as defined in claim 9 wherein said constant K is equal to 2a where a is the same coefficient a as in the Gaussian shaped pattern function G( xi ) e 